We want to apply Theorem REF .
For the subgroup \(K := \widehat{\mathop {\rm U{}}\nolimits }(\mathcal {H})\subseteq G := \widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})_0\) 
with Lie algebra
\({\mathfrak {k}}= {\mathfrak {z}}\oplus {\mathfrak {u}}(\mathcal {H})\) , we have
\( {\mathfrak {g}}_{\mathfrak {p}}_+ \oplus {\mathfrak {k}}_{\mathfrak {p}}_-,\quad \mbox{ where } \quad {\mathfrak {p}}_\pm = \ker (\mathop {{\rm ad}}\nolimits I \mp 2i\mathbf {1}) \) 
and \(I \in {\mathfrak {u}}(\mathcal {H})\)  is the multiplication with \(i\)  on \(\mathcal {H}\) .
Since the Lie algebra \({\mathfrak {k}}\)  of \(K\)  is complemented by the closed
subspace \({\mathfrak {p}}_2\)  of antilinear elements in \({\mathfrak {g}}\) , the coset space
\(G/K\)  carries the structure of a Banach homogeneous space
(cf. Lemma REF ).
Moreover, the closed \(K\) -invariant subalgebra
\({\mathfrak {k}}_{\mathfrak {p}}_+\)  determines on \(G/K\)  the structure of a
complex manifold for which the tangent space in the base point
can be identified with \({\mathfrak {g}}_({\mathfrak {k}}_ {\mathfrak {p}}_+) \cong {\mathfrak {p}}_-\) 
([1]}).
In view of (REF ), we obtain with
[2]} that the
group \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  acts transitively on the homogeneous
space \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H}_\beta _/P\) ,
where \(P\)  is the stabilizer of the subspace \(\mathcal {H}\oplus \lbrace 0\rbrace \subseteq \mathcal {H}_ (cf.\ Example~\ref {ex:liegrp}(d),(i)).From \) P Ores(H) = U(H)\( it now follows thatthe action of the Banach--Lie group \) G\( on \) G/K\( actually extends to aholomorphic action of a complex group.\) Let \(\Omega \in \Lambda ^0(\mathcal {H})\)  be the vacuum vector.
Then \(\Omega \)  is a smooth vector
which is an eigenvector for \(K\) . On \(\Lambda (\mathcal {H})\) , the operator
\(-i{\tt d}\pi (I)\)  is diagonal, and \(\Lambda ^n(\mathcal {H})\)  is the eigenspace
corresponding to the eigenvalue \(n\) . As \(-i\mathop {{\rm ad}}\nolimits I\)  acts on \({\mathfrak {p}}_-\)  with the
eigenvalue \(-2\) , it follows that
\({\tt d}\pi _a({\mathfrak {p}}_-)\Omega = \lbrace 0\rbrace ,\) 
so that Theorem REF  provides a holomorphic equivariant map
\( \eta \colon  G/K\rightarrow {\mathbb {P}}(\Lambda (\mathcal {H})^{\prime }), \quad g \mapsto \langle \cdot , \pi _a(g)^{-1} \Omega \rangle \) 
and a realization of \((\pi _a, \Lambda ^{\rm even}(\mathcal {H}))\) 
by holomorphic sections of a complex line bundle over \(G/K\) .
Since the representation
\((\pi _a, \Lambda ^{\rm even}(\mathcal {H}))\)  of \(G\) 
is irreducible, \(\Omega \)  is a cyclic vector.
As we have argued above, the action of every one-parameter subgroup of
\(G\)  extends to a holomorphic action of \( on \) G/K\(,so that Theorem~\ref {thm:5.7}(c) implies that\) I\( is the closed convex hull of the coadjoint orbitof \)\(. That \)\( has the desired form followsas in the proof of Theorem~\ref {thm:9.14}.\hfill \Box \)
Quasi-free representations
Let \(P = P^* = P^2\)  be an
orthogonal projection on \(\mathcal {H}\) ,
\(\mathcal {H}_- := \mathop {{\rm im}}\nolimits P\) , \(\mathcal {H}_+ := \ker P\) , and
\(\Gamma \colon  \mathcal {H}\rightarrow \mathcal {H}\)  be an isometric antilinear involution
commuting with \(P\) . Then
\(\tau _P := (\mathbf {1} - P) + \Gamma P \in \mathop {\rm O{}}\nolimits (\mathcal {H})\) .
We write \(a_0 \colon  \mathcal {H}\rightarrow B(\Lambda (\mathcal {H}))\)  for the map corresponding to
the Fock representation of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\) 
(Remark REF ).
Twisting with the Bogoliubov automorphism defined by \(\tau _P\) , we
obtain an irreducible
representation \((\pi _P, \Lambda (\mathcal {H}))\)  of \(\mathop {{\rm CAR}}\nolimits (\mathcal {H})\)  by
\( a_P(f) := a_0((\mathbf {1}-P)f) + a_0^*(\Gamma Pf)= \pi _0(a_{\tau _P}(f)) \quad \mbox{ for }\quad f \in \mathcal {H}, \) 
i.e., \(\pi _P = \pi _0 \circ \alpha _{\tau _P}\) .
These representations are called quasi-free.
For \(P =0\)  we recover the Fock representation defined by \(a_0\) .
Two quasi-free representations \(a_P\)  and \(a_Q\)  are equivalent if and only if
\(\Vert P - Q\Vert _2 < \infty \)  ([3]}).
Remark 10.13  The physical interpretation of \(P\)  is that its
range \(\mathcal {H}_-\)  consists
of the negative energy states and its kernel \(\mathcal {H}_+\)  consists of the
positive energy states.
For \(f \in \mathcal {H}_+\)  we have
\(a_P(f)\Omega = 0\)  and likewise
\(a_P^*(f)\Omega = 0\)  for \(f \in \mathcal {H}_-\) .
This is interpreted in such a way that the
creation operator \(a_P^*(f)\)  cannot create an additional negative
state from \(\Omega \)  because all negative states are already
filled (Pauli's Principle).
Likewise, the annihilation operator \(a_P(f)\) , corresponding
to the positive energy vector \(f\) , cannot extract any positive
energy state from \(\Omega \) .

The restricted unitary group \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
(cf. Example REF (c)) is a subgroup of \(\mathop {\rm O{}}\nolimits (\mathcal {H})\) ,
and for all projections \(Q = gPg^{-1}\) , \(g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
we have \(\Vert P - Q\Vert _2 < \infty \) , so that the equivalence of
\(a_P\)  and \(a_Q\)  leads to a projective unitary representation
of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H})\)  on \(\Lambda (\mathcal {H})\)  determined by
\( \pi _a^P(g) a_P(f) \pi _a^P(g)^* = a_P(gf), \quad f \in \mathcal {H},g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H}). \) 
Let \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H})\)  denote the corresponding central
extension and write \(\pi _a^P\)  for its unitary representation
on \(\Lambda (\mathcal {H})\) . To see that we thus obtain a semibounded
representation of a Lie group, we first note that
we have an embedding
\(\iota \colon  \mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {\rm O{}}\nolimits (\mathcal {H}), \quad g \mapsto \tau _P g \tau _P \) 
for which \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  is precisely the inverse image
of \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  and that for \(g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  we have
\(\pi _a(\tau _P g \tau _P)a_P(f)\pi _a(\tau _P g \tau _P)^*&= \pi _a(\tau _P g \tau _P)\pi _0(\alpha _{\tau _P} a(f))\pi _a(\tau _P g \tau _P)^*\\&= \pi _0(\alpha _{\tau _P g} a(f)) = \pi _P(a_g(f)) = a_P(gf),\) 

so that the projective representation of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\)  on
\(\Lambda (\mathcal {H})\)  coincide with \(\pi _a \circ \iota \) 
(cf. [4]}). It follows in particular that
\(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P) \cong \iota ^*\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
is a Lie group, \(\pi _a^P\)  is a smooth representation,
and since \(\mathop {{\rm im}}\nolimits (\mathop {\bf L{}}\nolimits (\iota ))\)  intersects the cone \(C_{{\mathfrak {u}}(\mathcal {H})}\subseteq W_{\pi _a}\) , the representation \(\pi _a^P\)  is semibounded
(cf. Proposition REF (iv)).
More precisely, each element \(x = (x_+, x_-) \in {\mathfrak {u}}(\mathcal {H}_+) \oplus {\mathfrak {u}}(\mathcal {H}_-)\)  with
\(i x_+ <\!\!< 0\)  and \(ix_- >\!\!> 0\)  lies in \(W_{\pi _a^P}\) .
This holds in particular for \(\tau _p I \tau _p = I(\mathbf {1} - 2P)\) .
In the physics literature, the
corresponding selfadjoint operator
\(Q := -i {\tt d}\pi _a^P(I) = {\tt d}\pi _a(\mathbf {1} - 2P)\) 
is called the charge operator and \(\mathbf {1} - 2P\)  the one-particle charge operator.
Its 1-eigenspace is \(\mathcal {H}_+\)  and its \(-1\) -eigenspace is \(\mathcal {H}_-\) .
Remark 10.14  The situation described above can be viewed as a
“second quantization” procedure that can be used to
turn a self-adjoint operator \(A\)  on the single particle space
\(\mathcal {H}\)  into a non-negative operator \(\widehat{A}\)  on the many particle
space. In fact, let \(P := P(]-\infty ,0[)\)  denote the
spectral projection of \(A\)  corresponding to the open negative axis.
Then \(iA\)  generates a strongly continuous one-parameter group
\(\gamma _A(t) := e^{itA}\) 
of \(\mathop {\rm U{}}\nolimits (\mathcal {H}_+) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_-) \subseteq \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) ,
and \(\pi _a^P(\gamma _A(t)) = e^{it \widehat{A}}\)  is a one-parameter group
of \(\mathop {\rm U{}}\nolimits (\Lambda (\mathcal {H}))\)  whose infinitesimal generator
\(\widehat{A}\)  has non-negative spectrum.

For more details on the complex manifolds
\(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\)  (the isotropic restricted Graßmannian)
and \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)/(\mathop {\rm U{}}\nolimits (P(\mathcal {H})) \times \mathop {\rm U{}}\nolimits (\mathcal {H}_+))\) 
(the restricted or Sato–Segal–Wilson Graßmannian) we refer to
[5]} and [6]}, where one also finds a detailed
discussion of the corresponding complex line bundles,
the Pfaffian line bundle \({\rm Pf}\)  over \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})/\mathop {\rm U{}}\nolimits (\mathcal {H})\) 
whose dual permits the even spin representation as a space of
holomorphic sections, and the determinant line bundle \({\rm Det}\) 
over the restricted Graßmannian whose dual provides a
realization of the representations of \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P)\) 
mentioned above. Physical aspects of highest weight representations
of \(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H})\)  are discussed in [7]}.

Perspectives
Classification problems:
In the preceding sections we mainly discussed three
prototypical classes of semibounded representations:
highest weight representations of the Virasoro group,
the metaplectic representation on the bosonic Fock space
and the spin representation on the fermionic Fock space.
These representations are of fundamental importance in
mathematical physics and homomorphisms from a Lie group
\(G\)  to \(\mathop {{\rm Sp}}\nolimits _{\rm res}(\mathcal {H})\) , \(\mathop {\rm O{}}\nolimits _{\rm res}(\mathcal {H})\)  or
\(\mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\) 
can be used to obtain semibounded representations of a central
extension by pulling back the representations discussed above.
This construction has been a major source of (projective) representations
for loop groups \(G = C^\infty ({\mathbb {S}}^1,K)\)  and \(\mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\) 
(cf. [8]}, [5]}, [10]}, [7]}).
What still remains to be developed is a better global perspective
on semibounded representations, including classification
results on irreducible ones and the existence of direct integral
decompositions.
Problem 11.1  Classify all irreducible semibounded unitary representations
of the groups \(\widehat{\mathop {{\rm Sp}}\nolimits }_{\rm res}(\mathcal {H})\) , \(\widehat{\mathop {\rm O{}}\nolimits }_{\rm res}(\mathcal {H})\) 
and \(\widehat{\mathop {\rm U{}}\nolimits }_{\rm res}(\mathcal {H},P)\) , where \(\mathcal {H}\)  is a complex Hilbert space and
\(P\)  an orthogonal projection on \(\mathcal {H}\) .
These classification problems are special cases of the more general
problem of the classification of the projective semibounded
unitary representations of the automorphism groups
of hermitian Hilbert symmetric spaces. For the three groups above,
the corresponding spaces of
\({\mathcal {I}}_\omega ^{\rm res}\) , \({\mathcal {I}}_\beta ^{\rm res}\) , resp.,
the restricted Graßmannian \(\mathop {{\rm Gr}}\nolimits _{\rm res}(\mathcal {H},P) :=\lbrace  gPg^{-1} \colon  g \in \mathop {\rm U{}}\nolimits _{\rm res}(\mathcal {H},P)\rbrace \) . A crucial
difference between these spaces is that the first one
is equivalent to a symmetric Hilbert domain
(of negative curvature) and the latter two are positively
curved spaces. This difference is also reflected in the difference
between the cones \(W_{\pi _s}\)  and \(W_{\pi _s}\) 
in \(\widehat{\mathfrak {sp}}_{\rm res}(\mathcal {H})\)  and \(\widehat{{\mathfrak {o}}}_{\rm res}(\mathcal {H})\) 
(Theorems REF  and REF ).
An even larger class of groups arises as automorphism groups
of Hilbert flag manifolds such as the orbit of a finite
flag in \(\mathcal {H}\)  under the group \(\mathop {\rm U{}}\nolimits _2(\mathcal {H})\) . For a systematic
discussion of these manifolds and the topology of the
corresponding real and complex groups we refer to
[2]} and for corresponding representations
to [13]}.

For loop groups of the form \(C^\infty ({\mathbb {S}}^1,K)\) ,
\(K\)  a compact connected Lie group, the irreducible
projective positive energy representations can be identified
as highest weight representations ([5]},
[15]}, and in particular
[8]} for \(K = ).With the convexity theorems in \cite {AP83} and \cite {KP84} it shouldbe possible to show that these representationsare semibounded and one can hope for an analog ofTheorem~\ref {thm:8.22} asserting that every irreduciblesemibounded representation of the corresponding double extensioneither is a highest weight representation or its dual.In \cite {Ne09b} it is shown on the algebraic level that therealso exist many interesting unitary representationsof infinite rank generalizations of twisted and untwistedloop algebras, resp., their double extensions(so-called locally affine Lie algebras). On the group levelthey correspond to (double extensions of) groupsof the form \) C(S1, K)\(, where\) K\( is a Hilbert--Lie group, such as \) U2(H)\(.It seems quite likely that all these representationsare semibounded with momentum sets generated by a singlecoadjoint orbit.\)New sources of semibounded representations:
Although many interesting classes of semibounded representations
are known, a systematic understanding of the geometric sources
of these representations is still lacking.
As we have seen above, on the Lie algebra level the existence
of open invariant cones in a trivial central extension
is necessary (Remark REF ). However, as the example
\(\mathcal {V}({\mathbb {S}}^1)\)  shows, it is not sufficient
(Theorem REF ).
Clearly, the circle \({\mathbb {S}}^1\)  plays a special role in many constructions,
as the rich theory for central extensions of loop groups and the
Virasoro group shows. Beyond \({\mathbb {S}}^1\) , it seems that
Lie algebras of conformal vector fields
(as generalizations of \(\mathcal {V}({\mathbb {S}}^1)\) ) (cf. Example REF , [17]} and [18]}) and Lie algebras of
sections of vector bundles over Lorentzian manifolds
(or more general “causal” spaces, [19]})
are natural candidates to be investigated
with respect to the existence of semibounded representations.
The latter class of Lie algebras is a natural
generalization of loop algebras.
Here an interesting point is that, although the conformal
groups \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^1) = \mathop {{\rm Diff}}\nolimits ({\mathbb {S}}^1)\)  of the circle and the conformal
group \(\mathop {\rm Conf{}}\nolimits ({\mathbb {S}}^{1,1})\)  of the Lorentzian torus
are infinite dimensional ,
for Lorentzian manifolds of dimension \(\ge 3\)  the conformal groups are finite
dimensional (cf. [20]}). In particular, it is contained in the list
of [17]}. This leads to the well-studied class of hermitian
Lie groups (see in particular [18]}, [23]}, [19]}).
Coadjoint orbits: There is a symplectic version of semibounded
representations, namely Hamiltonian actions
\(\sigma \colon  G \times M \rightarrow M\)  with a momentum map
\(\Phi \colon  M \rightarrow {\mathfrak {g}}^{\prime }\)  for which the image of the momentum map
is semi-equicontinuous. If \(G\)  is finite dimensional
and \(\Phi (M)\)  is closed, then Proposition REF 
implies that there exists an \(x \in {\mathfrak {g}}\)  for which the
Hamiltonian function \(H_x(m) := \Phi (m)(x)\)  is proper.
In particular, \(\Phi \)  is a proper map. In this sense
the semi-equicontinuity of the image of \(\Phi \)  is
a weakening of properness, which is a useful property
as far as convexity properties
are concerned ([25]}).
Even though we do not know in general to which extent coadjoint
orbits of infinite dimensional Lie groups are manifolds, the orbit
\(\mathcal {O}_\lambda \)  can always be viewed as the range of a momentum map
\(\Phi \colon  G \rightarrow {\mathfrak {g}}^{\prime }, g \mapsto \mathop {{\rm Ad}}\nolimits ^*(g)\lambda \)  corresponding to the
left action of \(G\)  on itself preserving the left invariant
closed 2-form \(\Omega \)  with \(\Omega _1(x,y) = \lambda ([x,y])\) .
This action is semibounded if and only if \(\mathcal {O}_\lambda \)  is
semi-equicontinuous.
Since the momentum sets of semibounded representations always consists
of semi-equicontinuous orbits, the identification of the set
\({\mathfrak {g}}^{\prime }_{\rm seq}\)  of semi-equicontinuous coadjoint orbits of a given
Lie algebra is already a solid first step towards the understanding
of corresponding semibounded representations.
As we have seen in many situations above, a useful tool to
study convexity properties of coadjoint orbits
are projection maps
\(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\) , where \({\mathfrak {t}}\subseteq {\mathfrak {g}}\)  is a
“compactly embedded” subalgebra, i.e.,
the action of \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {t}}}\)  on \({\mathfrak {g}}\)  factor through
the action of a compact abelian group.
As we have seen in Section , sometimes one
does not want to project to abelian subalgebras and
one has to study projections \(p_{\mathfrak {k}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {k}}\) ,
where \({\mathfrak {k}}\)  is a subalgebra for which \(e^{\mathop {{\rm ad}}\nolimits {\mathfrak {k}}}\) 
leaves a norm on \({\mathfrak {g}}\)  invariant
This is of particular interest to understand open invariant
cones \(W \subseteq {\mathfrak {g}}\)  because they often have the form
\( W = \mathop {{\rm Ad}}\nolimits (G) (W \cap {\mathfrak {k}}) \quad \mbox{ with } \quad p_{\mathfrak {k}}(W) = W \cap {\mathfrak {k}}\) 
(cf. Theorem REF ).
In this situation on needs convexity theorems of the type
\( p_{\mathfrak {k}}(\mathcal {O}_x) \subseteq \overline{\mathop {{\rm conv}}\nolimits }(\mathop {{\rm Ad}}\nolimits (N_G({\mathfrak {k}}))x) + C, \) 
where \(C\)  is a certain invariant convex cone in \({\mathfrak {k}}\)  and
\(N_G({\mathfrak {k}}) \subseteq G\)  is the normalizer of \({\mathfrak {k}}\)  in \(G\) 
(cf. [26]} for finite dimensional Lie algebras).
If \({\mathfrak {k}}= {\mathfrak {t}}\)  is abelian, then \(N_G({\mathfrak {t}})\)  is an analog
of the Weyl group.
For infinite dimensional Lie algebras, not many convexity theorems
are known. Of relevance for semibounded representations
is the particular case of affine Kac–Moody Lie algebras
([27]}, [28]}), and for Lie algebras of bounded
operators on Hilbert spaces
the infinite dimensional version of Kostant's Theorem by
A. Neumann ([29]}) is crucial. What is still lacking
is a uniform framework for results of this type.

Smooth vectors for representations
Let \(G\)  be a Lie group with Lie algebra \({\mathfrak {g}}\)  and exponential
function \(\exp _G \colon  {\mathfrak {g}}\rightarrow G\) .
Further, let \(V\)  be a locally convex space and
\(\pi \colon  G \rightarrow \mathop {{\rm GL}}\nolimits (V)\)  be a homomorphism
defining a continuous action of \(G\)  on \(V\) . We write
\(\pi ^v(g) := \pi (g)v\)  for the orbit maps and
\(V^\infty := \lbrace  v \in V \colon  \pi ^v \in C^\infty (G,V) \rbrace  \) 
for the space of smooth vectors.
In this appendix we collect some results of
[30]} that are used in the present paper.
Let
\({\tt d}\pi \colon  {\mathfrak {g}}\rightarrow \mathop {{\rm End}}\nolimits (V^\infty ), \quad {\tt d}\pi (x)v := \frac{d}{dt} \hbox{\vrule width0.5ptheight 5mm depth 3mm${{}\atop {{}\atop {\scriptstyle t=0}}}$} \pi (\exp tx)v \) 
denote the derived action of \({\mathfrak {g}}\)  on \(V^\infty \) .
That this is indeed a representation of \({\mathfrak {g}}\)  follows by observing that
the map \(V^\infty \rightarrow C^\infty (G,V), v \mapsto \pi ^v\) 
intertwines the action of \(G\)  with the right translation action on
\(C^\infty (G,V)\) , and this implies that the derived action
corresponds to the action of \({\mathfrak {g}}\)  on \(C^\infty (G,V)\)  by left invariant
vector fields (cf. [15]} for details).
Definition A.1  Let \(G\)  be a Banach–Lie group and
write \(\mathcal {P}(V)\)  for the set of continuous seminorms on \(V\) .
For each \(p \in \mathcal {P}(V)\)  and \(n \in {\mathbb {N}}_0\)  we define a seminorm
\(p_n\)  on \(V^\infty \)  by
\( p_n(v) := \sup \lbrace  p({\tt d}\pi (x_1) \cdots {\tt d}\pi (x_n)v) \colon \Vert x_i\Vert \le 1\rbrace  \) 
and endow \(V^\infty \)  with the locally convex topology defined by these
seminorms.

Theorem A.2  
If \((\pi , V)\)  is a representation of the Banach–Lie group
\(G\)  on the locally convex space \(V\)  defining a continuous action
of \(G\)  on \(V\) , then the
action \(\sigma (g,v) := \pi (g)v\)  of \(G\)  on \(V^\infty \)  is smooth.
If \(V\)  is a Banach space, then \(V^\infty \)  is complete,
i.e., a Fréchet space.

Theorem A.3  
If \((\pi , \mathcal {H})\)  is a unitary representation of a Lie group
\(G\) , then \(v \in \mathcal {H}\)  is a smooth vector if and only if the
corresponding matrix coefficient \(\pi ^{v,v}(g) := \langle \pi (g)v,v\rangle \) 
is smooth on a \(\mathbf {1}\) -neighborhood in \(G\) . If, in addition,
\(v\)  is cyclic, then the representation is smooth.

In the following we write \(\overline{u}\)  for the image of a unitary
operator \(u \in \mathop {\rm U{}}\nolimits (\mathcal {H})\)  in the projective unitary group
\(\mathop {\rm PU{}}\nolimits (\mathcal {H}) :=\mathop {\rm U{}}\nolimits (\mathcal {H})/\) .
Theorem A.4   Let \(G\)  be a connected Lie group,
\(\mathcal {H}\)  a complex Hilbert space and
\(\pi \colon  G \rightarrow \mathop {\rm U{}}\nolimits (\mathcal {H})\)  be a map with \(\pi (\mathbf {1}) = \mathbf {1}\) 
for which the corresponding map
\(\overline{\pi }\colon  G \rightarrow \mathop {\rm PU{}}\nolimits (\mathcal {H})\)  is a group homomorphism.
If there exists a \(v \in \mathcal {H}\)  for which
the function \((g,h) \mapsto \langle \pi (g)\pi (h)v, v\rangle \)  is
smooth on a neighborhood of \((\mathbf {1},\mathbf {1})\)  in \(G \times G\) ,
then the central
extension
\(\widehat{G} := \overline{\pi }^*\mathop {\rm U{}}\nolimits (\mathcal {H}) = \lbrace  (g,u) \in G \times \mathop {\rm U{}}\nolimits (\mathcal {H})\colon  \overline{\pi }(g) = \overline{u} \rbrace  \) 
of \(G\)  by \( is a Lie group and\) v\( is a smooth vector for therepresentation \) (, H)\( of \) G\( by\) (g,u) := u\(.\)
Proof.   
To exhibit \(\widehat{G}\)  as a Lie group, we have to show that
there exists a section \(\sigma \colon  G \rightarrow \widehat{G}\)  for which the
corresponding 2-cocycle
\( f_\sigma (g_1, g_2)= \sigma (g_1) \sigma (g_2) \sigma (g_1 g_2)^{-1} \) 
is smooth in a neighborhood of
\((\mathbf {1},\mathbf {1})\)  ([32]}).
Here we use that \(G\)  is connected.
A particular section \(\sigma \colon  G \rightarrow \widehat{G}\)  is given by
\(\sigma (g) = (g,\pi (g))\) .
Let \(U \subseteq G\)  be an open \(\mathbf {1}\) -neighborhood such that
\( \langle \pi (g)v,\pi (h)^{-1}v \rangle \ne 0 \quad \mbox{ for } \quad g,h \in U. \) 
Its existence follows from our continuity assumption.
If \(U^{\prime }\subseteq U\)  is an open \(\mathbf {1}\) -neighborhood with \(U^{\prime }U^{\prime } \subseteq U\) , we then
have for \(g_1, g_2 \in U^{\prime }\)
\( f_\sigma (g_1, g_2) \langle \pi (g_1 g_2)v, v\rangle = \langle \pi (g_1) \pi (g_2) v, v\rangle , \) 
which leads to
\( f_\sigma (g_1, g_2) = \frac{\langle \pi (g_1)\pi (g_2) v, v\rangle }{\langle \pi (g_1 g_2)v, v\rangle }. \) 
Therefore \(f_\sigma \)  is smooth in a neighborhood of \((\mathbf {1},\mathbf {1})\) .
This shows that \(\widehat{G}\)  is a Lie group and the multiplication map
\(G \times \widehat{G}, (g,t) \mapsto (g, t\pi (g))\) 
is smooth in a neighborhood of \(\mathbf {1}\)  ([32]}).
The representation
\(\widehat{\pi }\)  now satisfies
\( \langle \widehat{\pi }(g,t\pi (g))v, v \rangle = \langle t\pi (g)v, v \rangle = t \langle \pi (g)v,v\rangle , \) 
which is smooth in a neighborhood of \(\mathbf {1}\) . Now
Theorem REF  implies that \(v\)  is a smooth
vector for the representation \(\widehat{\pi }\) .
\(\Box \)
Remark A.5  The assumption that \(G\)  is connected in the preceding theorem
can be removed if \(G\)  is a Banach–Lie group. In this case we assume,
in addition, that \(\overline{\pi }\)  is continuous in the sense that
all functions \(g \mapsto |\langle \pi (g)v,w \rangle |\)  on \(G\)  are continuous
(cf. [34]}).
Then \(\widehat{G}\)  is the pullback of a central extension
of topological groups \(\mathop {\rm U{}}\nolimits (\mathcal {H}) \rightarrow \mathop {\rm PU{}}\nolimits (\mathcal {H})\) , hence in particular
a topological group with respect to the topology inherited from
\(G \times \mathop {\rm U{}}\nolimits (\mathcal {H})\) . Under the assumptions of Theorem REF ,
the central extension \(\widehat{G}_0\)  of the identity component
\(G_0\)  of \(G\)  carries a natural Lie group structure compatible
with the given topology on \(\widehat{G}\) . For each \(g \in \widehat{G}\) , the conjugation
map \(c_g\)  induces a continuous automorphism of the Banach–Lie group
\(\widehat{G}_0\) , and since continuous homomorphisms of Banach–Lie groups
are automatically smooth ([35]}),
\(c_g\)  also defines a Lie automorphism of
\(\widehat{G}_0\) . This implies that \(\widehat{G}\)  carries a unique Lie group
structure which coincides on the open subgroup \(\widehat{G}_0\)  with the given
one (see also [32]}).

Remark A.6  
In the situation of Theorem REF ,
the Lie algebra cocycle defining the central extension
\(\widehat{{\mathfrak {g}}}= \mathop {\bf L{}}\nolimits (\widehat{G})\)  of \({\mathfrak {g}}\)  by \({\mathbb {R}}\)  in the sense that
\( \widehat{{\mathfrak {g}}}= {\mathbb {R}}\oplus _{\eta } {\mathfrak {g}}, \quad [(t,x), (t^{\prime },x^{\prime })] = (\eta (x,x^{\prime }), [x,x^{\prime }]) \) 
can be calculated as follows.

If \({\tt d}\pi \colon  {\mathfrak {g}}\rightarrow \mathop {{\rm End}}\nolimits (\mathcal {H}^\infty )\)  is the map obtained
from the representation \({\tt d}\widehat{\pi }\)  via
\({\tt d}\pi (x) := {\tt d}\widehat{\pi }(0,x)\) , then we have for each unit
vector \(v \in \mathcal {H}^\infty \)  the relation
\( [{\tt d}\pi (x), {\tt d}\pi (y)]v= {\tt d}\pi ([x,y])v + i \eta (x,y)v, \) 
so that
\(\eta (x,y)&= \mathop {{\rm Im}}\nolimits \langle [{\tt d}\pi (x), {\tt d}\pi (y)]v, v\rangle - \mathop {{\rm Im}}\nolimits \langle {\tt d}\pi ([x,y])v, v \rangle \\&= 2\mathop {{\rm Im}}\nolimits \langle {\tt d}\pi (x)v, {\tt d}\pi (y) v\rangle + i \langle {\tt d}\pi ([x,y])v, v \rangle .\) 

The cone of positive definite forms on a Banach space
Let \(V\)  be a Banach space and
\(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  be the Banach space of continuous symmetric bilinear
maps \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\) , endowed with the norm
\(\Vert \beta \Vert  := \sup \lbrace  |\beta (v,v)| \colon  \Vert v\Vert  \le 1\rbrace . \) 
Clearly, the set \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  of positive semidefinite
bilinear maps is a closed convex cone in \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) .
Lemma B.1 

The cone \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  has interior points if and only if
\(V\)  is topologically isomorphic to a Hilbert space. If this is
the case, then its interior consists of all those positive definite
forms \(\beta \)  for which the norm \(\Vert v\Vert _\beta := \sqrt{\beta (v,v)}\) 
is equivalent to the norm on \(V\) .

Proof.   
Suppose first that \(V\)  carries a Hilbert space
structure \(\beta \) . Replacing the original norm by an equivalent
Hilbert norm, we may assume that \(V\)  is a real Hilbert space.
Then \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  can be identified with the space
\(\mathop {{\rm Sym}}\nolimits (V)\)  of symmetric operators on \(V\)  by assigning to
\(A \in \mathop {{\rm Sym}}\nolimits (V)\)  the form \(\beta _A(v,v) = \langle Av,w\rangle \) , satisfying
\( |\beta _A(v,v)| = |(Av,v)| \le \Vert A\Vert \Vert v\Vert ^2,\quad \mbox{ so that } \quad \Vert \beta _A\Vert  \le \Vert A\Vert .\) 
The polarization identity
\( (Av,w) = \beta _A(v,w) = \frac{1}{4} (\beta _A(v+w,v+w) -\beta _A(v-w,v-w)) \) 
further implies that \(\Vert A\Vert  \le 2 \beta _A\) , so that the
Banach spaces \(\mathop {{\rm Sym}}\nolimits (V)\)  and \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\)  are topologically isomorphic.
The identity \(\mathop {{\rm id}}\nolimits _V = \mathbf {1}\)  is an interior point in the cone
of positive operators, \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  has interior points.
Suppose, conversely, that \(\beta \in \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  is an
interior point. Then \(\beta - \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})_+\)  is a 0-neighborhood
in \(\mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) , which implies the existence of some
\(c > 0\)  such that \(\Vert \gamma \Vert  \le c\)  implies
\(\gamma (v,v) \le \beta (v,v)\)  for all \(v \in V\) .
Fixing \(v_0 \in V\) , we pick \(\alpha \in V^{\prime }\)  with
\(\Vert \alpha \Vert  = 1\)  and \(\alpha (v_0) = \Vert v_0\Vert \) .
Then \(\gamma (v,w) := c\alpha (v)\alpha (w)\)  is a symmetric bilinear form
with \(\Vert \gamma \Vert  = c\Vert \alpha \Vert ^2 = c\) . We therefore obtain
\( c \Vert v_0\Vert ^2 = \gamma (v_0, v_0) \le \beta (v_0, v_0)\le \Vert \beta \Vert  \Vert v_0\Vert ^2, \) 
showing that \(\beta \)  is positive definite and that the norm
\(\sqrt{\beta (v,v)}\)  is equivalent to the norm on \(V\) .
\(\Box \)
Remark B.2  
Let \((V,\omega )\)  be a weakly symplectic Banach space, i.e.,
\(\omega \)  is a non-degenerate skew-symmetric bilinear form.
Then
\( \mathfrak {sp}(V,\omega ) := \lbrace  X \in B(V) \colon  (\forall v,w\in V)\,\omega (Xv,w) + \omega (v,Xw) = 0\rbrace  \) 
is the corresponding symplectic Lie algebra. In particular,
\(X \in B(V)\)  belongs to \(\mathfrak {sp}(V,\omega )\)  if and only if
the bilinear form \(\omega (Xv,w)\)  is symmetric. The corresponding
quadratic function \(H_X(v) := {\textstyle {\frac{1}{2}}}\omega (Xv,v)\)  is called the
Hamiltonian function defined by \(X\).
If there exists an \(X \in \mathfrak {sp}(V,\omega )\)  for which
\((v,w) := \omega (Xv,w)\)  defines a Hilbert space structure
on \(V\) , then each continuous linear
functional \(\alpha \in V^{\prime }\)  is of the form \(i_v\omega \)  for
some \(v \in V\) . This means that
\((V,\omega )\)  is strongly symplectic,
i.e., the map
\(\Phi _\omega \colon  V \rightarrow V^{\prime }, v \mapsto i_v\omega \)  is surjective,
hence a topological isomorphism by the Open Mapping Theorem.
We further see that \(X\)  is injective. To see that it is also surjective,
let \(u \in V\)  and represent the continuous linear functional
\(i_u\omega \)  as \((w,\cdot )\)  for some \(w \in V\) . Then
\(\omega (u,v) = (w,v) = \omega (Xw,v)\)  for all \(v \in V\) ,
and thus \(Xw = u\) . Hence \(X\)  is a topological isomorphism,
satisfying \(\omega (u,v) = (X^{-1}u,v)\)  for \(v,w \in V\) . This further implies
that \(X\)  is skew-symmetric.

Proposition B.3  
Let \((V,\omega )\)  be a strongly symplectic Banach space.
Then the convex cone \(\lbrace  X \in \mathfrak {sp}(V,\omega ) \colon  H_X \ge 0\rbrace \)  has interior
points if and only if \(V\)  is topologically isomorphic to a
Hilbert space. If this is the case,
then there exists a complex structure \(I \in \mathfrak {sp}(V,\omega )\)  for
which \(H_I\)  defines a compatible Hilbert space structure on \(V\) , i.e.,
\( \omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w\rangle \) 
for the underlying complex Hilbert space.

Proof.   
(cf. [37]})
If \(\beta \colon  V \times V \rightarrow {\mathbb {R}}\)  is a symmetric bilinear form,
then \(\Phi _\beta \colon  V \rightarrow V^{\prime }, v \mapsto i_v\beta \)  also is a continuous
linear map, and
\(B := \Phi _\omega ^{-1} \circ \Phi _\beta \colon  V \rightarrow V\)  is continuous
linear with
\( \omega (Bv,v) = \Phi _\omega (Bv)(v)= \Phi _\beta (v)(v) = \beta (v,v), \) 
so that every symmetric bilinear form can be represented by an
element of \(\mathfrak {sp}(V,\omega )\) , and we obtain a topological isomorphism
\(\mathfrak {sp}(V,\omega ) \cong \mathop {{\rm Sym}}\nolimits ^2(V,{\mathbb {R}})\) . Combining this with
Lemma REF  proves the first assertion.
We now assume that \(V\)  is a real Hilbert space with the scalar product
\((v,w) = \omega (Xv,w)\) , where \(X\)  is as above. Then \(A := X^{-1}\)  is an
invertible skew-symmetric operator with \(\omega (v,w) = (Av,w)\) .
Then the complex linear extension \(A_ to the complex Hilbertspace \) V yields a self-adjoint operator \(iA_.The complex conjugation \)\( of \) V with respect to \(V\) 
now satisfies \(\sigma \circ iA_\sigma = - i A_, sothat \) Spec(iA\( is a compact symmetric subset of \) R\(, notcontaining \) 0\(. We therefore have an orthogonal decomposition\) V V+ V-\( into the positive and negative spectralsubspaces of \) iA. Since \(\sigma (V_\pm ) = V_\mp \) , we obtain
an isomorphism \(V \cong V_+\) , and hence a complex structure
\(I\)  on \(V\) , corresponding to multiplication by \(i\)  on \(V_+\) .
This means that \(v_+ \in V_+\)  with \(v = v_+ + \sigma (v_+)\) 
satisfies
\(\omega (Iv,v)&= (AIv,v)= (AI(v_+ + \sigma (v_+)), v_+ + \sigma (v_+)) \\&= (A_i v_+ - i \sigma (v_+)), v_+ + \sigma (v_+))= i (A_v_+ - \sigma (v_+)), v_+ + \sigma (v_+)) \\&= i \big ((A_v̏_+, v_+) - (v_+, A_v̏_+)\big )= 2 (iA_v̏_+, v_+),\) 

so we obtain a complex structure \(I\)  on \(V\) 
for which the Hamiltonian \(H_I\)  defines a Hilbert structure on
\(V\) . Since \(I\)  is skew-symmetric, \((V,I)\)  inherits the structure
of a complex Hilbert space with respect to the scalar product
\( \langle v, w \rangle := \omega (Iv,w) + i \omega (Iv,Iw) = \omega (Iv,w) + i \omega (v,w). \) 
Therefore our assumption leads to the representation
of \(\omega \)  as \(\omega (v,w) = \mathop {{\rm Im}}\nolimits \langle v, w \rangle \)  for a complex
Hilbert space structure on \(V\) .
\(\Box \)
Involutive Lie algebras with root decomposition
Definition C.1   (a) We call an abelian subalgebra
\({\mathfrak {t}}\)  of the real Lie algebra \({\mathfrak {g}}\)  a compactly embedded
Cartan subalgebra if \({\mathfrak {t}}\)  is maximal abelian and \(\mathop {{\rm ad}}\nolimits {\mathfrak {t}}\) 
is simultaneously diagonalizable on the complexification \({\mathfrak {g}}_with purely imaginary eigenvalues.Then we have a {\it root decomposition}\)\( {\mathfrak {g}}_ {\mathfrak {t}}_ \sum _{\alpha \in \Delta } {\mathfrak {g}}_\alpha , \)\(where \) g= { x g (h t [h,x]= (h)
x}\( and\)\(\Delta := \lbrace  \alpha \in {\mathfrak {t}}_* \setminus \lbrace 0\rbrace  \colon  {\mathfrak {g}}_\alpha \ne  \lbrace 0\rbrace \rbrace \)\( is the corresponding {\it root system}.\) If \(\sigma \colon  {\mathfrak {g}}_{\mathfrak {g}}_ denotes the complex conjugation withrespect to \) g\(, we write \) x* := -(x)\( for \) x g, so that
\({\mathfrak {g}}= \lbrace  x \in {\mathfrak {g}}_ x^* = -x\rbrace \) .
We then have
(I1)
\(\alpha (x) \in {\mathbb {R}}\)  for \(x \in i {\mathfrak {t}}\) .
(I2)
\(\sigma ({\mathfrak {g}}_\alpha ) = {\mathfrak {g}}_{-\alpha }\)  for \(\alpha \in \Delta .\)


Lemma C.2 
 For \(0 \ne  x_{\alpha } \in {\mathfrak {g}}_{\alpha }\)  the subalgebra
\({\mathfrak {g}}_x_\alpha ) := \mathop {{\rm span}}\nolimits \lbrace x_\alpha , x_{\alpha }^*,[x_\alpha , x_{\alpha }^*]\rbrace \) 
is \(\sigma \) -invariant and of one of the following types:
(A)
The abelian type: \([x_\alpha , x_{\alpha }^*] = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is two dimensional abelian.
(N)
The nilpotent type: \([x_\alpha , x_{\alpha }^*] \ne  0\) 
and \(\alpha ([x_\alpha , x_{\alpha }^*]) = 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha )\)  is a three dimensional Heisenberg algebra.
(S)
The simple type: \(\alpha ([x_\alpha , x_{\alpha }^*]) \ne  0\) ,
i.e., \({\mathfrak {g}}_x_\alpha ) \cong \mathop {{\mathfrak {sl} }}\nolimits _2(\) . In this case we distinguish the
two cases:
(CS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) > 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _2(\) , and
(NS)
\(\alpha ([x_\alpha , x_{\alpha }^*]) < 0\) , i.e.,
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\cong \mathop {{\mathfrak {su} }}\nolimits _{1,1}( \cong \mathop {{\mathfrak {sl} }}\nolimits _2({\mathbb {R}})\) .


Proof.   
First we note that, in view of \(x_\alpha ^* \in {\mathfrak {g}}_{-\alpha }\) ,
[38]} applies, and we see that \({\mathfrak {g}}_x_\alpha )\)  is of one of
the three types (A), (N) and (S). We note that
\(\alpha ([x_\alpha , x_\alpha ^*]) \in {\mathbb {R}}\) 
because of (I2) and \([x_\alpha , x_\alpha ^*] \in i{\mathfrak {t}}\) .
Now it is easy to check that
\({\mathfrak {g}}_x_\alpha ) \cap {\mathfrak {g}}\)  is of type (CS), resp., (NS), according to the sign
of this number.
\(\Box \)
The following proposition provides useful information for the
analysis of invariant cones and orbit projections.
Here we write \(p_{\mathfrak {t}}\colon  {\mathfrak {g}}\rightarrow {\mathfrak {t}}\)  for the
projection along \([{\mathfrak {t}},{\mathfrak {g}}] = \sum _{\alpha } ({\mathfrak {g}}_\alpha + {\mathfrak {g}}_{-\alpha }) \cap {\mathfrak {g}}\) ,
and \(p_{{\mathfrak {t}}^*} \colon  {\mathfrak {g}}^* \rightarrow {\mathfrak {t}}^*\)  for the restriction map.
Proposition C.3 

Let \(x \in {\mathfrak {t}}\) , \(x_\alpha \in {\mathfrak {g}}_\alpha \)  and
\(\lambda \in {\mathfrak {t}}^*\) . Then the following assertions hold:
(i)
\(p_{\mathfrak {t}}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits (x_\alpha - x_\alpha ^*)}x)= x +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \alpha (x) [x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] { \frac{\alpha (x)}{\alpha ([x_\alpha , x_\alpha ^*])}}[x_\alpha ^*, x_\alpha ] & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)
(ii)
\(p_{{\mathfrak {t}}^*}(e^{{\mathbb {R}}\mathop {{\rm ad}}\nolimits ^* (x_\alpha - x_\alpha ^*)}\lambda )= \lambda +{\left\lbrace \begin{array}{ll}{\mathbb {R}}^+ \lambda ([x_\alpha ^*, x_\alpha ]) \alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) \le 0 \\[0,2] \frac{\lambda ([x_\alpha ^*, x_\alpha ])}{\alpha ([x_\alpha , x_\alpha ^*])}\alpha & \text{ for }\ \alpha ([x_\alpha , x_\alpha ^*]) > 0.\end{array}\right.}\)


Proof.   
(i) is an immediate consequence of [26]},
and (ii) follows from (i) and the relation
\(p_{{\mathfrak {t}}^*}(\lambda )(e^{\mathop {{\rm ad}}\nolimits y}x)= \lambda (p_{\mathfrak {t}}(e^{\mathop {{\rm ad}}\nolimits y}x)).\) 
\(\Box \)
Some facts on Fock spaces
Let \(\mathcal {H}\)  be a complex Hilbert space. We endow the \(n\) -fold tensor
product with the canonical Hilbert structure defined by
\( \langle v_1 \otimes \cdots \otimes v_n, w_1 \otimes \cdots \otimes w_n \rangle := \prod _{j = 1}^n \langle v_j, w_j \rangle , \) 
and form the Hilbert space direct sum
\(T(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \mathcal {H}^{\otimes n}\) .
In \(\mathcal {H}^{\otimes n}\)  we write \(S^n(\mathcal {H})\)  for the closed subspace
generated by the symmetric tensors and
\(\Lambda ^n(\mathcal {H})\)  for the closed subspace generated by the
alternating tensors. We thus obtain subspaces
\( S(\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} S^n(\mathcal {H})\quad \mbox{ and } \quad \Lambda (\mathcal {H}) := \widehat{\bigoplus }_{n \in {\mathbb {N}}_0} \Lambda ^n(\mathcal {H}) \) 
of \(T(\mathcal {H})\)  and write \(P_s\) , resp., \(P_a\)  for the corresponding
orthogonal projections:
\( P_s(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)} \) 
and
\( P_a(f_1 \otimes \cdots \otimes f_n)= \frac{1}{n!} \sum _{\sigma \in S_n} \mathop {{\rm sgn}}\nolimits (\sigma )f_{\sigma (1)} \otimes \cdots \otimes f_{\sigma (n)}. \) 
The dense subspace \(S(\mathcal {H})_0 := \sum _{n \ge 0} S^n(\mathcal {H})\)  of \(S(\mathcal {H})\) 
carries an associative
algebra structure, given by
\( f_1 \vee \cdots \vee f_n := \sqrt{n!}P_s(f_1 \otimes \cdots \otimes f_n) \) 
and likewise \(\Lambda (\mathcal {H})_0 := \sum _{n \ge 0} \Lambda ^n(\mathcal {H})\) 
inherits an algebra structure defined by
\( f_1 \wedge \cdots \wedge f_n := \sqrt{n!}P_a(f_1 \otimes \cdots \otimes f_n). \) 
A unit vector \(\Omega \)  in the one-dimensional space
\(S^0(\mathcal {H}) = \Lambda ^0(\mathcal {H})\)  is called a vacuum vector.
Lemma D.1 
 We have
\(T \vee S= \sqrt{{n+m \atopwithdelims ()n}} P_s(T \otimes S) \quad \mbox{ for } \quad T \in S^n(\mathcal {H}), S \in S^m(\mathcal {H})\) 
and
\(T \wedge S= \sqrt{{n+m \atopwithdelims ()n}} P_a(T \otimes S) \quad \mbox{ for } \quad T \in \Lambda ^n(\mathcal {H}), S \in \Lambda ^m(\mathcal {H}).\) 


Proof.   
For the symmetric case we first note that
\(f^n = \sqrt{n!} f^{\otimes n}\) , so that
\( f^n \vee g^m= \sqrt{(n+m)!} P_s(f^{\otimes n} \otimes g^{\otimes m})= \sqrt{{n+m \atopwithdelims ()n}} P_s(f^n \otimes g^m). \) 
Since the elements \(f^n\)  generate \(S^n(\mathcal {H})\)  topologically,
(REF ) follows.
For the alternating case we obtain for
\(T = f_1 \wedge \cdots \wedge f_n\)  and
\(S = g_1 \wedge \cdots \wedge g_m\)  the relation
\(&\ \ \ \ \frac{1}{\sqrt{(n+m)!}}T \wedge S= P_a(f_1 \otimes \cdots \otimes g_m)\\&= P_a(P_a(f_1 \otimes \cdots \otimes f_n) \otimes P_a(g_1 \otimes \cdots \otimes g_m))= \frac{1}{\sqrt{n!m!}} P_a(T \otimes S).\) 

\(\Box \)
Remark D.2  
(a) For the norms of the product of \(T \in S^n(\mathcal {H})\)  and \(S \in S^m(\mathcal {H})\) ,
we obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \vee S\Vert = \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert P_s(T \otimes S)\Vert \le \frac{1}{\sqrt{n!}\sqrt{m!}} \Vert T \otimes S\Vert = \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}}.\) 
From that we derive for \(T \in S^k(\mathcal {H})\)  and \(n \in {\mathbb {N}}\)  the relation
\( \frac{1}{\sqrt{(kn)!}} \Vert T^n\Vert \le \Big (\frac{1}{\sqrt{k!}}\Vert T\Vert \Big )^n, \) 
and for \(T \in S^2(\mathcal {H})\)  we find in particular
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n},\) 
which converges for \(\Vert T\Vert ^2 < \frac{1}{2}\) .
(b) For \(T \in \Lambda ^n(\mathcal {H})\)  and \(S \in \Lambda ^m(\mathcal {H})\) 
we likewise obtain with (REF )
\( \frac{1}{\sqrt{(n+m)!}} \Vert T \wedge S\Vert \le \frac{\Vert T\Vert }{\sqrt{n!}} \frac{\Vert S\Vert }{\sqrt{m!}},\) 
which leads for \(T \in \Lambda ^2(\mathcal {H})\)  to
\(\Vert e^T\Vert ^2= \sum _{n = 0}^\infty \frac{1}{(n!)^2} \Vert T^n\Vert ^2\le \sum _{n = 0}^\infty \frac{(2n)!}{(n!)^2 2^n} \Vert T\Vert ^{2n}.\) 

Lemma D.3  
Let \(A\)  be an antilinear Hilbert–Schmidt operator on \(\mathcal {H}\) 
and define \(A^*\)  by \(\langle A^*v,w \rangle = \langle Aw,v\rangle \) .
If \(A^* = A\) , then there exists a unique element
\(\widehat{A} \in S^2(\mathcal {H})\)  with
\( \langle \widehat{A}, f_1 \vee f_2 \rangle = \langle Af_1, f_2 \rangle f\quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}, \) 
and if \(A^* = - A\) , there exists a unique element \(\widehat{A} \in \Lambda ^2(\mathcal {H})\) 
with
\( \langle \widehat{A}, f_1 \wedge f_2 \rangle = \langle Af_1, f_2 \rangle \quad \mbox{ for } \quad f_1, f_2 \in \mathcal {H}. \) 
Moreover, the following assertions hold:
(i)
\(\Vert \widehat{A}\Vert ^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2\) .
(ii)
If \(A\)  and \(B\)  are hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = {\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .
(iii)
If \(A\)  and \(B\)  are skew-hermitian and antilinear, then
\(\langle \widehat{A}, \widehat{B} \rangle = -{\textstyle {\frac{1}{2}}}\mathop {{\rm tr}}\nolimits (AB)\) .


Proof.   
(a) First we consider the case where \(A^* = A\) .
Let \((e_j)_{j \in J}\)  be an orthonormal basis of \(\mathcal {H}\) .
Then we have in \(S^2(\mathcal {H})\)  the relations
\( \Vert e_j^2\Vert ^2 = 2\Vert  e_j \otimes e_j\Vert ^2 = 2\quad \mbox{ and } \quad \Vert e_j \vee e_k\Vert ^2= 2\Vert {\textstyle {\frac{1}{2}}}(e_j \otimes e_k + e_k \otimes e_j)\Vert ^2 = 1, \quad j\ne k. \) 
If \(<\)  denotes a linear order on \(J\) , we thus obtain the
orthonormal basis
\(\frac{1}{\sqrt{2}} e_j^2, \quad e_j \vee e_k, \quad j < k, \quad \mbox{ of } \quad S^2(\mathcal {H}).\) 
This leads to
\(\Vert \widehat{A}\Vert ^2&= \sum _j {\textstyle {\frac{1}{2}}}|\langle \widehat{A}, e_j^2 \rangle |^2+ \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2 \\&= \sum _j {\textstyle {\frac{1}{2}}}|\langle Ae_j, e_j \rangle |^2+ \sum _{j<k} |\langle Ae_j, e_k \rangle |^2= {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
For \(A^* = -A\)  we similarly get
\(\Vert \widehat{A}\Vert ^2&= \sum _{j<k} |\langle \widehat{A}, e_j \vee e_k \rangle |^2= \sum _{j<k} |\langle Ae_j, e_k \rangle |^2 = {\textstyle {\frac{1}{2}}}\Vert A\Vert _2^2.\) 
(b) Let \(\mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\)  denote the complex subspace
of hermitian antilinear
Hilbert–Schmidt operators.
Then the prescription \(\langle A, B \rangle := \mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\)  defines a
sesquilinear form on this space with
\( \overline{\mathop {{\rm tr}}\nolimits (AB)} = \mathop {{\rm tr}}\nolimits ((AB)^*) = \mathop {{\rm tr}}\nolimits (B^*A^*) = \mathop {{\rm tr}}\nolimits (BA), \) 
so that it is hermitian. Its restriction to the diagonal
satisfies
\(\langle A,A \rangle = \mathop {{\rm tr}}\nolimits (A^2)= \sum _{j \in J} \langle A^2e_j, e_j \rangle = \sum _{j \in J} \langle Ae_j, Ae_j \rangle = \Vert A\Vert _2^2. \) 
In view of (a), polarization implies that
\(\mathop {{\rm tr}}\nolimits (AB) = \langle A, B \rangle = 2 \langle \widehat{A}, \widehat{B}\rangle \) 
for \(A, B \in \mathop {{\rm Herm}}\nolimits _2(\mathcal {H})_a\) .
(c) For the space \(\mathop {{\rm Aherm}}\nolimits _2(\mathcal {H})_a\)  of skew-hermitian antilinear
Hilbert–Schmidt operators the same argument works with
\(\langle A, B \rangle := -\mathop {{\rm tr}}\nolimits (AB) = \mathop {{\rm tr}}\nolimits (AB^*)\) .
\(\Box \)
